[en] Quantum Phase Transitions (QPT) are phase transitions that occur as a function of a coupling constant in the quantum Hamiltonian describing the system. Atomic nuclei provide examples of quantum shape phase transitions. The phases of the system are different geometric structures of nuclei characterized by different dynamic symmetries, U(5), U(3) and SO(6), corresponding to spherical, axially deformed and so-called γ-unstable shapes. The shape phase diagram of nuclei is discussed. It is shown that the transition from spherical, U(5), to axially deformed, U(3), is first order, between spherical and γ-unstable, SO(6), is second order, and between axially deformed and γ-unstable is a crossover. Critical exponents and scaling behavior, i.e. dependence on the number of particles, N, are also discussed. Recently, it has been found that at the critical value of a QPT, a new symmetry emerges, called ''critical symmetry'', and related to scale invariance. In the second part of this talk, this (unexpected) symmetry is discussed and experimental examples shown. Finally, the role of supersymmetry in phase transitions in mixed systems of bosons and fermions will be mentioned. QPT in nuclei provide one of the best experimentally verified examples of phase transitions in physics.

[en] The role of dynamic symmetries and supersymmetries in nuclei is reviewed. The concept of critical symmetry, appropriate to describe bosonic systems (even-even nuclei) at the critical point of a phase transition, is introduced, and the symmetry, E(5), at the critical point of spherical to γ-unstable shape phase transition, is discussed. The recently introduced concept of critical supersymmetry, appropriate to describe mixed systems of bosons and fermions (odd-even nuclei) at the critical point of a phase transition is presented. The case of a j=3/2 particle at the critical point of spherical to γ-unstable transition, called E(5/4), is discussed

[en] A brief description of the role of symmetry in phase transitions is given. Quantum phase transitions (QPT) in algebraic models are discussed, both at the semi-classical and quantal level. The newly introduced concept of excited states quantum phase transitions (ESQPT) is presented.

[en] The role of symmetry in (shape) quantum phase transitions (QPT) in nuclei is briefly reviewed. Finite size scaling is discussed. The concepts of excited state quantum phase transitions (ESQPT) and of critical matter are introduced. The ab initio theory of QPT is mentioned.

[en] The method of spectrum generating algebras (SGA) and dynamic symmetries (DS) for the space part of the hadronic wave function is briefly reviewed. Some results for mesons and baryons are presented

[en] Some challenges faced by nuclear structure physics in the next decade are briefly mentioned. Particularly important are, in experimental nuclear structure physics, the study and understanding of quantum shape phase transitions in mixed Bose-Fermi systems and in two-fluid systems, and, in theoretical nuclear structure physics, the development of up-to-date theories of collective motion.

[en] Three open problems in neutrino physics are mentioned. A brief review of neutrino- less double beta decay (DBD) is given. The question of quenching of g_A is discussed. Other possible scenarios, Majoron emission, sterile neutrinos and non-standard mechanisms, are presented and their implication to DBD discussed. (paper)

[en] The role of discrete symmetries in nuclear physics is briefly reviewed within the context of the algebraic cluster model (ACM). The symmetries D_3 (triangle) for 3α and T_d (tetrahedron) for 4α are discussed and evidence shown for their occurrence in "1"2C (D_3) and "1"6O (T_d)

[en] Quantum phase transitions (QPT) and excited state quantum phase transitions (ESQPT) in algebraic models are briefly discussed, in particular those occurring in the U(3) model of bending vibrations of molecules. Experimental evidence in the water molecule, H₂O, is presented. (paper)

[en] An overview of the method of spectrum generating algebras (SGA) and dynamical symmetries (DS) is given and applied to a class of models (the so-called s-b boson models) with SGA≡u(n). Quantum phase transitions (QPT) in these systems are discussed by introducing the coset spaces U(n)/U(n-1)⊗U(1). Finally, spectrum generating superalgebras (SGSA) and dynamical supersymmetries (SUSY) of a class of Bose-Fermi systems with SGA≡u(n/m) are introduced and applied to s-b Bose-Fermi models. (paper)